Abstract not found

There are 105212 Carmichael numbers up to 10 : we describe the calculations.

Motivated by the discovery that the eighth root of the theta series of the E8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f ∈ R (where R = 1 + xZ[[x]]) can be written as f = g n for g ∈ R, n ≥ 2. Let Pn: = {g n | g ∈ R} and let µn: = n ∏ p|n p. We show among other things that (i) for f ∈ R, f ∈ Pn ⇔ f (mod µn) ∈ Pn, and (ii) if f ∈ Pn, there is a unique g ∈ Pn with coefficients mod µn/n such that f ≡ gn (mod µn). In particular, if f ≡ 1 (mod µn) then f ∈ Pn. The latter assertion implies that the theta series of any extremal even unimodular lattice in Rn (e.g. E8 in R8) is in Pn if n is of the form 2i3j5k (i ≥ 3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the rth order Reed-Muller code of length 2m is in P2r (and similarly that the theta series of the Barnes-Wall lattice BW2m is in P2m). We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f ∈ Pn (n ≥ 2) with coefficients restricted to the set {1, 2,..., n}.

We extend the method due originally to Loh and Niebuhr for the generation of Carmichael numbers with a large number of prime factors to other classes of pseudoprimes, such as Williams's pseudoprimes and elliptic pseudoprimes. We exhibit also some new Dickson pseudoprimes as well as superstrong Dickson pseudoprimes.

Euler's totient function f has the property that f(n) is the order of the group U(n) of units in Z n (the integers mod n). In the early years of the twentieth century, Carmichael defined a similar function l, where l(n) is the exponent of U(n). He called an element of U(n) with order l(n) a primitive l-root of n.

Dedicated to the memory of Prof. Arnold Ross

This paper studies the mapping g x → g x2 (mod p) by showing that it inherits its structure from the mapping x → x 2 (mod ordp(g)). We first explore the context in which this mapping is important; specifically, we consider the Diffie-Hellman protocol, Diffie-Hellman problem, and the various problems that have spawned from these including the discrete logarithm and square exponent problems. We then demonstrate that we may infer the structure of g x → g x2 (mod p) from x → x 2 (mod ordp(g)) and translate some results proved by Somer and Kroger about the latter to the former case.

Abstract. Let ϕ and λ be the Euler and Carmichael functions, respectively. In this paper, we establish lower and upper bounds for the number of positive integers n ≤ x such that ϕ(λ(n)) = λ(ϕ(n)). We also study the normal order of the function ϕ(λ(n))/λ(ϕ(n)). 1

Abstract. Bounds and other relations involving variables connected with Carmichael numbers are reviewed and extended. Families of numbers or individual numbers attaining or approaching these bounds are given. A new algorithm for finding three-prime Carmichael numbers is described, with its implementation up to 10 24. Statistics relevant to the distribution of threeprime Carmichael numbers are given, with particular reference to the conjecture of Granville and Pomerance in [10]. 1.